The Birch and Swinnerton-Dyer conjecture implies Goldfeld's conjecture
Alexander Smith
TL;DR
The paper proves that Birch and Swinnerton-Dyer (BSD) in the quadratic twist family of an elliptic curve $E/Q$ implies Goldfeld’s conjecture for that family, by showing that exactly half the twists have $2^{ nullfty}$-Selmer corank $0$ and half have corank $1$. It develops a generalized framework for higher Selmer ranks in twist grids, including balanced isogeny analyses (Cases IV/V), uses Tamagawa ratios and the Cassels–Tate pairing to control isogeny Selmer behavior, and computes refined moments of $2$-Selmer groups on grids of twists. Through a Sawin–Matchett-Wood style category argument, these moments determine a unique distribution for $2$-Selmer ranks, which then lift to the $2^{ nullfty}$-Selmer corank distribution. Consequently, BSD in the twist family forces Goldfeld’s conjecture for $E$, with corank parity aligned to root numbers and a precise 50/50 distribution between corank $0$ and $1$ twists. The results also clarify the distinct distributions from the Poonen–Rains model in certain twist families and extend prior work to curves with balanced isogenies in Case IV/V.
Abstract
Given an elliptic curve E/Q, we show that 50% of the quadratic twists of E have $2^{\infty}$-Selmer corank 0 and 50% have $2^{\infty}$-Selmer corank 1. As one consequence, we prove that the Birch and Swinnerton-Dyer conjecture implies Goldfeld's conjecture. Previously, this result was known by work of the author for elliptic curves over Q satisfying certain technical conditions. As part of this work, we determine the distribution of 2-Selmer ranks in the quadratic twist family of E. In the cases where this distribution was not already known, it is distinct from the model for distributions of 2-Selmer groups constructed by Poonen and Rains.